{"id":56555,"date":"2022-07-12T12:34:37","date_gmt":"2022-07-12T15:34:37","guid":{"rendered":"https:\/\/militares.estrategia.com\/portal\/?p=56555"},"modified":"2022-11-06T10:31:49","modified_gmt":"2022-11-06T13:31:49","slug":"progressao-geometrica","status":"publish","type":"post","link":"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/progressao-geometrica\/","title":{"rendered":"Progress\u00e3o Geom\u00e9trica (PG): o que \u00e9, propriedades e mais!"},"content":{"rendered":"\n<p>Estudando para entrar nas escolas militares? Confira os resumos que o Estrat\u00e9gia Militares preparou sobre os assuntos mais cobrados nos seletivos! Contamos a voc\u00ea<strong> o que \u00e9 uma Progress\u00e3o Geom\u00e9trica (PG), suas propriedades e muito mais.\u00a0<\/strong><\/p>\n\n\n\n<p>Antes de come\u00e7armos, que tal <strong>assinar nossa Newsletter<\/strong>? Trazemos semanalmente as principais not\u00edcias do mundo dos concursos militares, dicas de carreira e curiosidades!&nbsp;<\/p>\n\n\n\n<p><\/p>\n\n\n<div id=\"new-form\">\n\t<form id=\"forms_layout\">\n\t<input type=\"hidden\" id=\"chave_de\" name=\"chave_de\" value=\"01_EMIL_LEADS_TOTAIS\" \/>\n\t\t<input type=\"hidden\" id=\"mid\" name=\"mid\" value=\"515009017\" \/>\n\t\t<input type=\"hidden\" id=\"Objetivo\" name=\"Objetivo\" value=\"Militares\" \/>\n\t\t<input type=\"hidden\" id=\"Estado_de_Origem_do_IP\" name=\"Estado_de_Origem_do_IP\"  value=\"\" \/>\n\t\t<input type=\"hidden\" id=\"Cidade_de_Origem_do_IP\" name=\"Cidade_de_Origem_do_IP\" value=\"\" \/>\n\t\t<input type=\"hidden\" id=\"Modo_de_entrada\" name=\"Modo_de_entrada\" value=\"Portal\" \/>\n\t\t<input type=\"hidden\" id=\"Pagina_Origem\" name=\"Pagina_Origem\" value=\"\" \/>\n\n\t\t<div class=\"texto\">\n\t\t\t<h3 class=\"news\">Inscreva-se em nossa newsletter!<\/h3>\n\t\t\t<p>Receba not\u00edcias sobre os mais 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Privacidade<\/a> e aceita receber informa\u00e7\u00f5es adicionais do Estrat\u00e9gia Educacional.\n\t\t\t<input id=\"checkbox_layout\" type=\"checkbox\" checked=\"checked\" required \/>\n\t\t\t<span class=\"checkbox\"><\/span>\n\t\t<\/label>\n\t\t<div id=\"msg\"><\/div>\n\t\t<div id=\"error\" class=\"error\">\n\t\t<\/div>\n\t<\/form>\n<\/div><!-- \/new-form -->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_79_2 counter-hierarchy ez-toc-counter ez-toc-transparent ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">Navegue pelo conte\u00fado\t<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/progressao-geometrica\/#O-que-sao-Sequencias\" >O que s\u00e3o Sequ\u00eancias?&nbsp;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/progressao-geometrica\/#O-que-e-uma-Progressao-Geometrica\" >O que \u00e9 uma Progress\u00e3o Geom\u00e9trica?&nbsp;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/progressao-geometrica\/#Como-encontrar-o-Termo-Geral-da-PG\" >Como encontrar o Termo Geral da PG?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/progressao-geometrica\/#Quais-as-propriedades-de-uma-PG\" >Quais as propriedades de uma PG?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/progressao-geometrica\/#Progressao-Aritmetica-Geometrica-PAG\" >Progress\u00e3o Aritm\u00e9tica Geom\u00e9trica (PAG)&nbsp;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/progressao-geometrica\/#Hora-de-praticar\" >Hora de praticar!<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/progressao-geometrica\/#Veja-tambem\" >Veja tamb\u00e9m:<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-o-que-sao-sequencias\"><span class=\"ez-toc-section\" id=\"O-que-sao-Sequencias\"><\/span>O que s\u00e3o Sequ\u00eancias?&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Antes de definir a Progress\u00e3o Geom\u00e9trica, tenha em mente que ela \u00e9 <strong>um tipo de sequ\u00eancia \u2013 uma s\u00e9rie de n\u00fameros naturais que correspondem a um n\u00famero real ou complexo<\/strong>. Existem alguns tipos delas, sendo que os mais conhecidos s\u00e3o a pr\u00f3pria PG e a Progress\u00e3o Aritm\u00e9tica (PA).&nbsp;<\/p>\n\n\n\n<p>Em uma sequ\u00eancia, o termo \u00e9 dado como ?<sub>n<\/sub>, sendo que <em>n<\/em> equivale \u00e0 posi\u00e7\u00e3o do termo e \u00e9 sempre um n\u00famero natural. De forma que a nota\u00e7\u00e3o da sequ\u00eancia se apresenta da seguinte forma:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">(?<sub>1<\/sub>,?<sub>2<\/sub>,?<sub>3<\/sub>,\u2026,?<sub>n<\/sub>)<\/p>\n\n\n\n<p>As sequ\u00eancias seguem uma l\u00f3gica, chamada de<strong> Lei de Forma\u00e7\u00e3o.<\/strong> Ela permite calcular qualquer termo de uma sequ\u00eancia levando em conta a posi\u00e7\u00e3o dele, ou seja, o n.<\/p>\n\n\n\n<pre class=\"wp-block-verse has-electric-grass-gradient-background has-background\">Para entender mais sobre o que s\u00e3o Sequ\u00eancias e a PA, visite <a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/o-que-e-pa\/\" target=\"_blank\" >\u201cProgress\u00e3o Aritm\u00e9tica: o que \u00e9, tipos e propriedades\u201d<\/a>.&nbsp;<\/pre>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-o-que-e-uma-progressao-geometrica\"><span class=\"ez-toc-section\" id=\"O-que-e-uma-Progressao-Geometrica\"><\/span>O que \u00e9 uma Progress\u00e3o Geom\u00e9trica?&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Uma Progress\u00e3o Geom\u00e9trica \u00e9 uma sequ\u00eancia com a seguinte <strong>Lei de Forma\u00e7\u00e3o:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub> = ?<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub> = ?<sub>?\u2212?<\/sub>.?<\/p>\n\n\n\n<p>Em outras palavras, voc\u00ea sempre <strong>multiplica o termo anterior por um mesmo valor para obter o n\u00famero seguinte da sequ\u00eancia<\/strong>. Esse valor recebe o nome de <em>raz\u00e3o<\/em> e \u00e9 representado por <em>q<\/em>.&nbsp;<\/p>\n\n\n\n<p>Confuso? Confira alguns exemplos:&nbsp;<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>(1, 3, 9, 27, \u2026). PG cujo primeiro termo \u00e9 ?<sub>1<\/sub> = 1 e sua raz\u00e3o \u00e9:<\/li><\/ol>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/0r97z49Cjhn8jURKBhQ-tr2KcNBJFi6_NUHiDNs8JKy4hsqjxBNimduu7F-0uhemTi0r80ylXfg6ib790DXVXBkaKHB_Wuacr6QkuIHCCFpIL3kZfFeZOJqelUNes0LLMTHQtpFoS0R-zEHbmg\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\"><li><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/F8ufC6JqX_4XTGg052ukqwtBM01VK-gqKPb7XBxUVjuXkaV4xt86p4rmUWU4ozdVgj1xdQD-KmJ73GKcORyzFV4yb6_2qANlk1uC-BhD9bwQNK00z54KBcOp7o6i5szKN_rec3I21i97XdicsA\" width=\"75\" height=\"38.09177264629121\">. PG com ?<sub>1<\/sub> = 1 e raz\u00e3o:<\/li><\/ol>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/VrSzOuZyqSBDxC9TYg9fnFnyFj622UsJ7VwVcIEiqfXFW1NYclpzlsrgZE02iZ4VFfOE5SD_O5u1wgxOQxfnbOQZQqwF-TFtLxu2FzgKIzGUivO8ZWZGpOZQNNTjRopwhvlZNLHJv9G2WhHAuA\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>3) (3, \u2212 9, 27, \u2212 81,\u2026). PG com ?<sub>1<\/sub> = 3 e raz\u00e3o:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/evLf19d5Q8RN47R3NX7YtFlnVuEcsFjB8zbG6YtfHff1zCJFj-vt8BRmWUXXaghfzVYoXjHXP6mmS9DjELAYOOeSObOEsY09MUXAS1Jp7eR-f8cAxUEGDiSrZdOk1Y8ugJV9aoMYavWjPRl8gA\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-como-encontrar-o-termo-geral-da-pg\"><span class=\"ez-toc-section\" id=\"Como-encontrar-o-Termo-Geral-da-PG\"><\/span>Como encontrar o Termo Geral da PG?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Se voc\u00ea quiser encontrar um termo da PG pela posi\u00e7\u00e3o que ele ocupa, <strong>voc\u00ea pode calcular o termo geral da PG por meio da f\u00f3rmula de recorr\u00eancia.<\/strong> Para entender o racioc\u00ednio, escreva os termos da seguinte forma:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>2<\/sub> = ?<strong><sub>1<\/sub><\/strong>.?<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>3<\/sub> = ?<strong><sub>2<\/sub><\/strong>.?<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>4<\/sub> = ?<strong><sub>3<\/sub><\/strong>.?<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>5<\/sub> = ?<strong><sub>4<\/sub><\/strong>.?<\/p>\n\n\n\n<p class=\"has-text-align-center\">\u22ee<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub> = ?<strong><sub>?\u22121<\/sub><\/strong>.?<\/p>\n\n\n\n<p>Perceba que temos <strong>? \u2212 1 <\/strong>termos. Vamos multiplic\u00e1-los:&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/22_4rGC6vV-cA_P_-AYpCMIjftdxKlWWNd8bqPcMsIqSmVDzcWHEzGwbMgMb7LqOeAOzViTQW1Vi4n4ldAFPnRepiVxkMa9J5N1Z4gg6yMaJYQa2lBebjTgnPkNS50s9ZX3LlUsNq0CZUGAHpw\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Os termos em vermelho se cancelam e assim obtemos o termo geral da PG:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>?<\/strong><strong><sub>?<\/sub><\/strong><strong> = ?<\/strong><strong><sub>1<\/sub><\/strong><strong>.?<\/strong><strong><sup>?\u22121<\/sup><\/strong><\/p>\n\n\n\n<p>Tamb\u00e9m podemos escrever o termo geral em fun\u00e7\u00e3o de um termo qualquer da PG. Usando o termo geral, podemos escrever:<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub> = ?<sub>1<\/sub>.?<sup>?\u22121<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub> = ?<sub>1<\/sub>.?<sup>?\u22121<\/sup><\/p>\n\n\n\n<p>Dividindo as duas equa\u00e7\u00f5es:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/7jHcWvjVvq4N5xrQw3dPMMCurL-GXerYcT9J0Fz7IL5wcwwKA96e3KCv1bJaonJ98uxOXXoBiS5xj_FtclgaIKMOf70XF7TMiDE8LtHQlDsjRrFOligI3g21oJeC4d8PX8NTMNj2Up6HJ-A7Og\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub> = ?<sub>?<\/sub>.?<sup> (?\u22121) \u2212(?\u22121)<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub> = ?<sub>?<\/sub>. ? <sup>(?\u2212?)<\/sup><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-quais-as-propriedades-de-uma-pg\"><span class=\"ez-toc-section\" id=\"Quais-as-propriedades-de-uma-PG\"><\/span>Quais as propriedades de uma PG?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Assim como a PA, a PG tamb\u00e9m conta com algumas propriedades que facilitam na hora de calcular os resultados procurados.&nbsp;<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-termos-equidistantes\">Termos Equidistantes<\/h3>\n\n\n\n<p>A primeira propriedade aponta que <strong>o produto dos termos equidistantes \u00e9 um valor constante e a soma dos \u00edndices \u00e9 igual \u00e0 ? + 1.<\/strong> Para prov\u00e1-la, vamos escrever os termos equidistantes generalizados ?<sub>?+1<\/sub> e ?<sub>?\u2212?<\/sub>, ?\u2208\u2115, usando o termo geral da PG.<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?+1<\/sub> = ?<sub>1<\/sub>. ? <sup>(?+1) \u22121<\/sup> = ?<sub>1<\/sub>. ?<sup>?<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?\u2212?<\/sub> = ?<sub>1<\/sub>. ? <sup>(?\u2212?) \u22121<\/sup><\/p>\n\n\n\n<p>Multiplicando os dois termos, temos:<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?+1<\/sub>.?<sub>?\u2212?<\/sub> = (?<sub>1<\/sub>.?<sup>?<\/sup>) (?<sub>1<\/sub> ? <sup>(?\u2212?) \u22121<\/sup>)<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?+1<\/sub> .?<sub>?\u2212?<\/sub> = ?<sub>1<\/sub> .?<sub>1<\/sub> .?<sup>?<\/sup> .? <sup>[(?\u2212?) \u22121]<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?+1<\/sub> .?<sub>?\u2212?<\/sub> = ?<sub>1<\/sub> .?<sub>1<\/sub> .?<sup>?\u22121<\/sup><\/p>\n\n\n\n<p>Sabemos que ?<sub>?<\/sub> = ?<sub>1<\/sub>. ?<sup>?\u22121<\/sup>, substituindo na equa\u00e7\u00e3o acima:<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?+1<\/sub> .?<sub>?\u2212?<\/sub> = ?<sub>1<\/sub> .?<sub>?<\/sub><\/p>\n\n\n\n<p>Logo, o produto dos termos equidistantes \u00e9 igual ao valor do produto dos extremos.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-soma-dos-termos-de-pg-finita\">Soma dos termos de PG finita<\/h3>\n\n\n\n<p>Se uma PG for<strong> finita<\/strong>, \u00e9 poss\u00edvel <strong>calcular a soma de todos os seus termos <\/strong>por meio da seguinte f\u00f3rmula:&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/HtXyXoJ22vnbq_O3uihMGlwKTvl2Zgw0Q4vm0NvjU9tGwPChz9wRgWf5R3oYSwOdWGGlrjgIaJOJQy-CylmMwSrya-Q0NhmA1jJ8SdZ-kPHHCPzqVRkNiRZwHLK3XKbSoq_3Ypv3kNO1RJUMUA\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Para chegar nela \u00e9 preciso seguir o racioc\u00ednio a seguir. Primeiro, escreva a soma da PG finita:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub> = ?<sub>1<\/sub> + ?<sub>2<\/sub> + ?<sub>3<\/sub> + \u2026 + ?<sub>?\u22121<\/sub> + ?<sub>?<\/sub><\/p>\n\n\n\n<p>Se usar o termo geral para escrever os termos:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>2<\/sub> = ?<sub>1<\/sub>.?<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>3<\/sub> = ?<sub>1<\/sub> .?<sup>2<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>4<\/sub> = ?<sub>1<\/sub>. ?<sup>3<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">\u22ee<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub> = ?<sub>1<\/sub> .?<sup>?\u22121<\/sup><\/p>\n\n\n\n<p>Agora, substitu\u00edmos os termos em S<sub>n<\/sub>:<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub> = ?<sub>1<\/sub> + ?<sub>1<\/sub>? + ?<sub>1<\/sub> .?<sup>2<\/sup> + ?<sub>1<\/sub> .?<sup>3<\/sup> + \u2026 + ?<sub>1<\/sub> .?<sup>?\u22121<\/sup><\/p>\n\n\n\n<p>Multiplicando ?<sub>?<\/sub> por ?, obtemos:<\/p>\n\n\n\n<p class=\"has-text-align-center\">?.?<sub>?<\/sub> = ?<sub>1.<\/sub>? + ?<sub>1<\/sub> ?<sup>2<\/sup> + ?<sub>1<\/sub> ?<sup>3<\/sup> + ?<sub>1<\/sub>. ?<sup>4<\/sup> + \u2026 + ?<sub>1<\/sub> ?<sup>?<\/sup><\/p>\n\n\n\n<p>Subtraindo as duas equa\u00e7\u00f5es:<\/p>\n\n\n\n<p class=\"has-text-align-center has-small-font-size\">?<sub>?<\/sub> \u2212 ?.?<sub>?<\/sub> = ?<sub>1<\/sub> +<strong> ?<\/strong><strong><sub>1<\/sub><\/strong><strong>? + ?<\/strong><strong><sub>1<\/sub><\/strong><strong> ?<\/strong><strong><sup>2<\/sup><\/strong><strong> + ?<\/strong><strong><sub>1<\/sub><\/strong><strong> ?<\/strong><strong><sup>3<\/sup><\/strong><strong> + \u2026 + ?<\/strong><strong><sub>1<\/sub><\/strong><strong> ?<\/strong><strong><sup>?\u22121<\/sup><\/strong><strong> \u2212<\/strong> (<strong>?<\/strong><strong><sub>1<\/sub><\/strong><strong>? + ?<\/strong><strong><sub>1<\/sub><\/strong><strong> ?<\/strong><strong><sup>2<\/sup><\/strong><strong> + ?<\/strong><strong><sub>1<\/sub><\/strong><strong> ?<\/strong><strong><sup>3<\/sup><\/strong><strong> + ?<\/strong><strong><sub>1<\/sub><\/strong><strong> ?<\/strong><strong><sup>4<\/sup><\/strong><strong> + \u2026 +<\/strong> ?<sub>1<\/sub> .?<sup>?<\/sup>)<\/p>\n\n\n\n<p>Repare que os termos em negrito se cancelam. Dessa forma:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub> \u2212 ?.?<sub>?<\/sub> = ?<sub>1<\/sub> \u2212 ?<sub>1<\/sub> .?<sup>?<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub> (1 \u2212 ?) = ?<sub>1<\/sub> (1 \u2212 ?<sup>?<\/sup>)<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/snp8WBbcisaK6n6V0ouTfH4dPb1Hj1Y_HGbfE0WBiAki2WA25bK9krBzHuHJOPX__g9QRRAU9IAkMW3OWzyo4nmmxRksXMly49ErRuWglZkOGCHz1STsRpQvw9g1VIHoU6wavr6I0KHV6fbGz88\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Tamb\u00e9m podemos escrever:&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/l6B62mrc_XoGg_etC8aNMq7eXjLHB0p55rwz2UMjCQB2TfQpKBiVFi-a8SEb6oCdRC7Sw_aIL8O6MR02uK6fwn9oiMOm6uc1-UJQJqatk2Vbvpd5PR0cjoimpHE-vYoGGUL7frbixwz5NxbSJWA\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-soma-dos-termos-de-pg-infinita\">Soma dos termos de PG infinita<\/h3>\n\n\n\n<p>Se a PG for<strong> infinita,<\/strong> no entanto, e a raz\u00e3o absoluta for menor que 1, a soma dos seus termos converge para a<sub>1<\/sub>\/(1-q), como demonstrado na f\u00f3rmula a seguir, com 0&lt;|q|&lt;1:&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"73\" height=\"42\" src=\"https:\/\/militares.estrategia.com\/portal\/wp-content\/uploads\/2022\/07\/image-1.png\" alt=\"\" class=\"wp-image-56557\"\/><\/figure><\/div>\n\n\n\n<p>Observe a demonstra\u00e7\u00e3o:&nbsp;<\/p>\n\n\n\n<p>(?<sub>1<\/sub>,?<sub>2<\/sub>,?<sub>3<\/sub>,\u2026) ?? ????????<\/p>\n\n\n\n<p>Vamos escrever a f\u00f3rmula da soma da PG finita:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/aNORX0Dc7tMJyI6HuftTbYzJBo9Cnu6k3o9Dn3fS3datj7OXO0QhdDctNjHRi7iGvyYzSMxJ2ckvxxcrTgxzHb1ZfICeXSDuzmtT78hjcgeJS3Og-UmsERMPq3sZe27oYp5eto7qKmID2z7j6RE\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Se \u2212 1 &lt; ? &lt; 1 e a PG \u00e9 infinita podemos escrever:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/NvamnXxGemwgFtLNmV43euVGvV4OpPTuQF1SwZrmDygp4zOqcBBU_VTob1NtD3taioOSdatxt-rendf_QkgTjZphpVDtUzNcsefmdD9l_3tPXB079sryosu-ruSpeIr6uLX1cMwRwg1pFCAjXhI\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Ainda est\u00e1 confuso? Observe a sequ\u00eancia a seguir:&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/m-XvdKPTWXDu0V9DoFlNjdqJ-y9a76VzkMHA_4jMDsk0BQNC6nyTZ_lvpP8dP9D3pcsIOqS7pABO_2FvlzccSkUxAOkOR2Z9e5tqVqRs6NI4xzLfiQgwfx7NHqHSDE9KnoEB7knWUcSO7KKha8Q\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Perceba que os valores v\u00e3o diminuindo \u00e0 medida que o \u00edndice aumenta. Vamos calcular ?<sub>10<\/sub>:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/xkeVPtLGzaeh3-VOfT7yGcRB8XgbmeXT1D-aIpHY0Ijh4hvMIuFolYqy7Y2wE6TgslW2FOOl1zpROCZTcbcrSLz69X5yYRrhE_RsvvzSYLw-KB0TgjMhitELjVpmkivlq3KXeC-cuI_VP8EYyTo\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Se tentarmos calcular ?<sub>?<\/sub> tal que ? seja t\u00e3o grande, vamos <strong>obter um valor muito pr\u00f3ximo<\/strong> <strong>de zero.<\/strong> Assim, quando ? tende ao infinito, podemos escrever:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/0oJaz9flStdTnrxm3kObCmxDZJesxWKOEWsYk_M49l91EbGGPdn-8BPy33MTbQnp7gYp9HUFIJwoQ1WMPoVSUb5frRLi8HnDRjgyE24y17XaOTlBC0mI9iT3ZxMw5jIN_CLq5gvJTPRvFQ1R9Bs\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Essa propriedade \u00e9 v\u00e1lida para n\u00fameros pertencentes ao intervalo ] \u2212 1, 1[.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-produto-dos-termos-da-pg\">Produto dos termos da PG<\/h3>\n\n\n\n<p>Outra propriedade \u00e9 a multiplica\u00e7\u00e3o dos termos da PG. A f\u00f3rmula \u00e9:&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/AjyYlfgEttrB6656v5dDMbsDqAiMRaGTW89NFKr8-BZs0dY0frYkaIZR723zi9Qk5HKcD15I6gdxB2JRy9mENLMmpeafxE2XQKLgZ4PIIHE3WIARoEsEuAgZE_KoWKEUGod6Or5UyOilEhXAlhE\" alt=\"\" width=\"124\" height=\"56\"\/><\/figure><\/div>\n\n\n\n<p>Entenda como ela \u00e9 obtida! Primeiro, escreva os termos da PG usando o termo geral:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>1<\/sub> = ?<sub>1<\/sub><\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>2<\/sub> = ?<sub>1<\/sub>.?<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>3<\/sub> = ?<sub>1<\/sub>. ?<sub>2<\/sub><\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>4<\/sub> = ?<sub>1<\/sub> ?<sub>3<\/sub><\/p>\n\n\n\n<p class=\"has-text-align-center\">\u22ee<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub> = ?<sub>1<\/sub> .?<sub>?\u22121<\/sub><\/p>\n\n\n\n<p>Multiplicando os termos da PG, obtemos:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/NRE6mtqEjvM11hFPH8FdOURVf--kS_VZvlmon02lkdgXUOfppe1I_TwjUsEK5pTAWdOpItXD5Gr2UdVLe0N8HnD3TXi6GyBlaFtUSoiOg1DjuQhRl8ryyJEg1CZLKYDiYdogwfBbh7VTKfN0yQ\" alt=\"\" width=\"380\" height=\"66\"\/><\/figure><\/div>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub> = ?<sub>1<\/sub><sup>?<\/sup> ?<sup>1+2+3+..+ (?\u22121)<\/sup><\/p>\n\n\n\n<p>1 + 2 + 3 + \u2026 + (? \u2212 1) \u00e9 uma soma de PA de raz\u00e3o 1. Aplicando a f\u00f3rmula da soma da PA:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/dGwlwKXC_WnX8ZXiF2oKh4EDqwG44l5oUUYUYNlnOyTbfBEUeDcMBYskbNqAh8QHvPhMffmwirZa-xs-ZoRe2SHUVFWL8o6IHjQz9fwzpfWuGLvvY4pVFK7QKp01YXWPXXlCYhghv12nwF0H5Q\" alt=\"\" width=\"268\" height=\"43\"\/><\/figure><\/div>\n\n\n\n<p>Substituindo em ?<sub>?<\/sub>:&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/4oO47XiZ6dRj0P0Lx8uHu9WDIvaT_GKSYpbC5wAZLDWzscYg9L1mKiSR53G3suZZ-1mstNjTf92UI7ggcqbONKWADftyvDUqCav2v2Q3ATgI9F0MzMrShkzvRqm1FpYs6LTLr7LvfkMrq--2hA\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p><strong>No caso de <\/strong><strong><em>n<\/em><\/strong><strong> ser \u00edmpar, <\/strong>podemos escrever tamb\u00e9m:&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/YBDTs_ONPHVrAtTSEst_FEPmLNUccIqS4eoxb0Nwg0xCWXAz8soDxnIKtwyXyrKOLQUXxIK_8taS9bLXz7CipFySDKiR26xlmEivie3oA9GZ1JDbkSt8kzt8HzZNrkbzQ9OA-7150xO05sxuhA\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Note que <img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/o2txaIFx1x4YUfF79RtigLeGh6YqaVsAEo9gJCXiejjhSwJItHwfuuZugcpCHYHgtlHZGrh4Di9ii29ylrUeE5oE-eJ--Kp1EDTMvRTLuWAJ9iA4DmZT7iqtZ6xhWq1xDEiAP60Nb0WZVyY3tQ\" width=\"36\" height=\"35\"> \u00e9 o termo m\u00e9dio da PG. Para provar essa afirma\u00e7\u00e3o, fazemos da seguinte forma:&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/UU2WxNAoQ18w208xbjgtZR5BVfAottKGvHSDTQf0VLP5uTKN9rn3IVgeEaHiKVQYghHWhlX2ksojvyrTIu7GQJyslm-AmTwtoAseLvGfxDDxYBlHwVQPHqacP8TfbDpjAy311b0bEGc98NtlIQ\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Pela f\u00f3rmula dos produtos dos termos da PG, temos:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/4d_LglnSZSiVHrmCfMr_Is7Zcbw5p0lecXMd9gH_uGZHf4i6VoGPf_WXIHPnUTP8srCJGDGlR3zoKP3bLqd5DaB-uILw-VFne4naRAge-eRaNXTouw2s-DLWWpfNBT43QajAHJX2xjx1cr6DeA\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"h-tome-nota\">Tome nota!&nbsp;<\/h4>\n\n\n\n<p>O<strong> produto dos termos de uma sequ\u00eancia<\/strong> tamb\u00e9m pode ser representado dessa forma:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/WCZxsMFgXJji_X8jLJt2KbuaK2m73gI0v7qp8dyDOLcY2OrUyuAkwKzylFOh5-9xu1Po6c25RVQDE0aPYmX-r_UM3m6CCRPM9a5rT09v1oQYspNh4rrCl5eubfqzq3E4MXmmoeuczWuvAgZLPw\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>\u03a0 \u00e9 o s\u00edmbolo usado para <strong>representar o produt\u00f3rio de uma sequ\u00eancia<\/strong>. Ela parte do termo de \u00edndice ? = 1 e vai at\u00e9 o termo de \u00edndice ?.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-media-geometrica\">M\u00e9dia Geom\u00e9trica<\/h3>\n\n\n\n<p>Por meio da M\u00e9dia Geom\u00e9trica conseguimos expressar os termos sem usar a raz\u00e3o. Como assim? Representando ?<sub>?\u22121<\/sub> e ?<sub>?+1<\/sub> usando o termo geral da PG:<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?\u22121<\/sub> = ?<sub>1<\/sub> .? <sup>(?\u22121) \u22121<\/sup> = ?<sub>1<\/sub> .?<sup>?\u22122<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?+1<\/sub> = ?<sub>1<\/sub> .? <sup>(?+1) \u22121<\/sup> = ?<sub>1<\/sub> .?<sup>?<\/sup><\/p>\n\n\n\n<p>Multiplicando os termos, obtemos:<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?\u22121<\/sub> .?<sub>?+1<\/sub> = ?<sup>1<\/sup> .?<sup>?\u22122<\/sup> .?<sub>1<\/sub> .?<sup>?<\/sup> = ?<sub>1<\/sub><sup>2<\/sup> .?<sup>2?\u22122<\/sup> = ?<sub>1<\/sub><sup>2<\/sup> ?<sup>2(?\u22121)<\/sup> = [?<sub>1<\/sub> .?<sup>?\u22121<\/sup>] <sup>2<\/sup> = ?<sub>?<\/sub><sup>2<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">\u21d2 ?<sub>?\u22121<\/sub> .?<sub>?+1<\/sub> = ?<sub>?<\/sub><sup>2<\/sup><\/p>\n\n\n\n<p>Assim, para uma PG (?<sub>1<\/sub>,?<sub>2<\/sub>,?<sub>3<\/sub>), podemos escrever ?<sub>2<\/sub> como a m\u00e9dia geom\u00e9trica de ?<sub>1<\/sub> e ?<sub>3<\/sub>:<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub><sup>?<\/sup> = ?<sub>?<\/sub> .?<sub>?<\/sub><\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-notacao-especial\">Nota\u00e7\u00e3o Especial<\/h3>\n\n\n\n<p>Outra forma de escrever uma PG \u00e9 da seguinte forma:&nbsp;<\/p>\n\n\n\n<p><strong>PG com 3 termos:<\/strong>&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/1uToghME8tckfVNNSRmQQFyIWj2Ebdva5Mrfs5oFEiEYWLLt2RheBrLaoJwvUeLarEKaqd3GnYhXlXmeRPRVZJ-MTjhZWYkSpmWBZhM50_hms8lCGx7DbweVTfUh9vjjB2M0FENyVci8yrH3Yw\" alt=\"\" width=\"168\" height=\"45\"\/><\/figure><\/div>\n\n\n\n<p><strong>PG com 4 termos:&nbsp;<\/strong><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/IZFsAzsdr9AdD2PnxWM8x0kHsTbVjQ3SyBWBYqVWP0XH38bcyGH7D1MAJr8gW6PN-NcVohbbbT1GrqMqEvWM4KeUy7-7vHgpUWBCeY8qw1KawoL69TuaC_ZB41KgfUoDzFDmIx4Kausmn23atA\" alt=\"\" width=\"243\" height=\"48\"\/><\/figure><\/div>\n\n\n\n<p><strong>PG com 5 termos:&nbsp;<\/strong><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/00pA8tpNJKdcv4MhltFOBx4RO1261_PbkuiaCf4UL4m2wCpBKSfefQMGSQx3AzzUAUyBJ4GWU_A22uyzhpIpoba_QOX1Z11EhPDOfmtTfozODw3rZfH0Z1SfRYKcGAL-xX0afGdXRb9vRx5pHQ\" alt=\"\" width=\"271\" height=\"37\"\/><\/figure><\/div>\n\n\n\n<p>Note que a raz\u00e3o da PG com 4 termos \u00e9 ? = ?<sup>2<\/sup>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-progressao-aritmetica-geometrica-pag\"><span class=\"ez-toc-section\" id=\"Progressao-Aritmetica-Geometrica-PAG\"><\/span>Progress\u00e3o Aritm\u00e9tica Geom\u00e9trica (PAG)&nbsp;<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Entre as sequ\u00eancias, tamb\u00e9m podemos ter uma que \u00e9 a<strong> uni\u00e3o de uma PA com uma PG. <\/strong>Essa sequ\u00eancia chama-se progress\u00e3o aritm\u00e9tica geom\u00e9trica (PAG). A defini\u00e7\u00e3o de seu termo geral \u00e9 dada por:&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub> = [?<sub>?<\/sub> + (? \u2212 ?) ?] ?<sup>?\u2212?<\/sup><\/p>\n\n\n\n<p>Veja os exemplos:&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/pISgY_wBDdgJ3tTokfHKMWp7YamRV81BQ0z3iGjx37aYuN9GNnt2jfx3fnpu5tRf32DXqDHwPGS3HV_wD_be3sLzu451iHd1dU97BKSQDdv9N3t0kF9U8k8W5xBZppun_X7UXeUkFM1-xmtmHw\" alt=\"\"\/><\/figure>\n\n\n\n<p>Perceba que o numerador \u00e9 uma PA de raz\u00e3o 1 (1, 2, 3, 4, 5,\u2026) e o denominador \u00e9 uma PG de raz\u00e3o 2 (1, 2, 4, 8, 16,\u2026) . Essa sequ\u00eancia \u00e9 uma PAG.<\/p>\n\n\n\n<p>Para a PAG, a raz\u00e3o ? \u00e9 igual a 1 e a raz\u00e3o ? \u00e9 igual a 1\/2. O termo geral \u00e9:<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>?<\/sub> = [?<sub>1<\/sub> + (? \u2212 1) ?] ?<sup>?\u22121<\/sup><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/uZs6IuIBqSc63UBzk-BNQ0sc_pKVYcWJ4uFEhFnAux5E19KdpX85wrF543giGhF87eRslnHIiNMXJqUH_2ym6khTAnBCQL6-ZOJBYoRa9DVD6m-iYj5-qBeRcL93bYj14MvSl3hV3gm8E1COXQ\" alt=\"\" width=\"290\" height=\"53\"\/><\/figure><\/div>\n\n\n\n<p>As quest\u00f5es que podem cair na prova sobre PAG normalmente<strong> cobrar\u00e3o a soma dos termos da PAG. <\/strong>Vamos aprender a resolver esse tipo de quest\u00e3o.<\/p>\n\n\n\n<p>Considere o termo inicial da PAG ?<sub>1<\/sub> = ? e raz\u00f5es ? e ?. Vamos calcular:<\/p>\n\n\n\n<p class=\"has-text-align-center\">? = ? + [(? + ?) ?] + [(? + 2?) ?<sup>2<\/sup>] + [(? + 3?) ?<sup>3<\/sup>] + \u2026<\/p>\n\n\n\n<p>O bizu dessa quest\u00e3o \u00e9 multiplicar ? por ? e fazer ? \u2212 ??. Veja:<\/p>\n\n\n\n<p class=\"has-text-align-center\">?? = ?? + [(? + ?) ?<sup>2<\/sup>] + [(? + 2?) ?<sup>3<\/sup>] + [(? + 3?) ?<sup>4<\/sup>] + \u2026<\/p>\n\n\n\n<p>Fazendo ? \u2212 ??:<\/p>\n\n\n\n<p class=\"has-text-align-center\">? \u2212 ?? = ? + [<strong>(? + ?) ?<\/strong>] + [(<em>? + 2?) ?<\/em><em><sup>2<\/sup><\/em>] + [(? + 3?) ?<sup>3<\/sup>] + \u2026<\/p>\n\n\n\n<p class=\"has-text-align-center\">\u2212 {<strong>??<\/strong> + [<em>(? + ?) ?<\/em><em><sup>2<\/sup><\/em>] + [(? + 2?) ?<sup>3<\/sup>] + [(? + 3?) ?<sup>4<\/sup> + \u2026}<\/p>\n\n\n\n<p>Repare que os termos em negrito, it\u00e1lico e sublinhados de forma semelhante podem ser subtra\u00eddos. Dessa forma, obtemos:<\/p>\n\n\n\n<p class=\"has-text-align-center\">? (1 \u2212 ?) = ? + ?? + ??<sup>2<\/sup> + ??<sup>3<\/sup> + ??<sup>4<\/sup> + \u2026<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/8l_qvISZxsYmqwwK5e6D-1z3Mcz2Fqs3iOtln1ce8kBr8Sw7kaGs-ACEFuibZisEkIQvwtaDuX3dmxKIIli_Tec5kLsaPSAm4aeiL7G3SM19ZzuOqOpk2QL3-NDaz36bK-mJkB4a5vX2-t9qWA\" alt=\"\" width=\"239\" height=\"70\"\/><\/figure><\/div>\n\n\n\n<p>Encontramos uma f\u00f3rmula para a soma de uma PAG de raz\u00e3o \u2212 1 &lt; ? &lt; 1.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-hora-de-praticar\"><span class=\"ez-toc-section\" id=\"Hora-de-praticar\"><\/span>Hora de praticar!<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Com o conhecimento b\u00e1sico de PG em mente, que tal testar o que aprendeu? Resolva o exerc\u00edcio abaixo:&nbsp;<\/p>\n\n\n\n<p><em>Dado a PG (?, \u2212 ?, ?, \u2212 ??, ??,\u2026), calcule:<\/em><\/p>\n\n\n\n<p><em>a) ???<\/em><\/p>\n\n\n\n<p><em>b) ???<\/em><\/p>\n\n\n\n<p><strong>Resolu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<p>a) Para calcular os termos da PG, precisamos encontrar sua raz\u00e3o. Podemos aplicar a f\u00f3rmula de recorr\u00eancia:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/PNgeQ5jeVpMRk60bwtjtzLyUHl9JSw4UIo2YVQEIrTAAAbXVTsrUCY2MRc7_pJQabxZKx822uPaFZpy5NykDjwgMtvfD0uxr0RUEcgCLPr-5BxW6pK4N5i8xpCHRyuMTTG3z1WQTB2IotRIDew\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Observando a sequ\u00eancia, ?<sub>2<\/sub> = \u2212 4 e ?<sub>1<\/sub> = 2. Substituindo na f\u00f3rmula acima:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/C-S8ieLa4CXp5pyDhOdRVY-nfymHcn3NsNjwY32d2TRjfBlED9TqvJeIupXq9dUxIVAVNfeZsAVYO0B57OM3XKl4zlvVd8onEi3Jct0hw7fZP7Eigtlp_ei9DVhMqQTc2cV9PgXDj75SRjSW7Q\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Usando o termo geral para calcular ?<sub>20<\/sub>:<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>20<\/sub> = ?<sub>1<\/sub> .?<sup>20\u22121<\/sup> = ?<sub>1<\/sub> .?<sup>19<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>20<\/sub> = 2 (\u2212 2) <sup>19<\/sup> = \u2212 2<sup>20<\/sup><\/p>\n\n\n\n<p>b) Conhecemos a raz\u00e3o e o termo inicial, vamos aplicar o termo geral:<\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>30<\/sub> = ?<sub>1<\/sub> .?<sup>30\u22121<\/sup> = ?<sub>1<\/sub> .?<sup>29<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">?<sub>30<\/sub> = 2 (\u2212 2) <sup>29<\/sup> = \u2212 2<sup>30<\/sup><\/p>\n\n\n\n<p>Gostou do conte\u00fado? Acompanhe o Portal do Estrat\u00e9gia Militares e tenha <a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/\" target=\"_blank\" >acesso a mais resumos dos nossos professores<\/a>! Al\u00e9m disso, se quiser treinar ainda mais seus conhecimentos, assine o Banco de Quest\u00f5es do Estrat\u00e9gia Militares!\u00a0<\/p>\n\n\n\n<p>S\u00e3o milhares de quest\u00f5es dos mais variados n\u00edveis, tanto in\u00e9ditas quanto de concursos anteriores, para voc\u00ea aproveitar! Clique no banner e saiba mais!<\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><strong><a id=\"cta\" class=\"cta-imagem\" href=\"https:\/\/militares.estrategia.com\/curso\/banco-de-questes-anual\/\" target=\"blank\">\n                <img decoding=\"async\" width=\"100%\" src=\"https:\/\/militares.estrategia.com\/portal\/wp-content\/uploads\/2023\/02\/EM-Banco-de-Questoes.jpg\" alt=\"EM - Banco de Questo\u0303es\" title=\"EM - Banco de Questo\u0303es\">\n        <\/a><\/strong><\/p>\n\n\n\n\n\n<p><\/p>\n\n\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Veja-tambem\"><\/span>Veja tamb\u00e9m:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<ul class=\"wp-block-list\"><li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/o-que-e-pa\/\">Progress\u00e3o Aritm\u00e9tica (PA): o que \u00e9, tipos e propriedades<\/a><\/li><li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/par-ordenado-e-produto-cartesiano\/\">Par ordenado e produto cartesiano: o que s\u00e3o e suas propriedades<\/a><\/li><li><a href=\"https:\/\/militares.estrategia.com\/portal\/materias-e-dicas\/matematica\/relacao-binaria\/\">O que \u00e9 uma Rela\u00e7\u00e3o Bin\u00e1ria na Matem\u00e1tica<\/a><\/li><\/ul>\n","protected":false},"excerpt":{"rendered":"Conte\u00fado Did\u00e1tico elaborado pelo professor Victor So\n","protected":false},"author":9,"featured_media":56560,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"wl_entities_gutenberg":"","footnotes":""},"categories":[307],"tags":[],"wl_entity_type":[187],"class_list":{"0":"post-56555","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-matematica","8":"wl_entity_type-article"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.6 (Yoast SEO v26.6) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Progress\u00e3o Geom\u00e9trica (PG): o que \u00e9, propriedades e mais!<\/title>\n<meta name=\"description\" content=\"Voc\u00ea sabe o que \u00e9 uma Progress\u00e3o Geom\u00e9trica (PG)? 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